Former Senator David Perdue of Georgia has decided he will not run against an incumbent Democrat, Senator Raphael Warnock, in 2022, just a week after Mr. Perdue announced he had filed paperwork to kick-start a possible new campaign.Mr. Perdue, a Republican and former businessman who lost in a January runoff election to the state’s other newly elected senator, Jon Ossoff, said in a statement that he had reached the decision after “much prayer and reflection” with his wife, Bonnie.Mr. Warnock defeated Kelly Loeffler, who was also a Republican incumbent, in January, winning a term that expires in January 2023. The two defeats deadlocked Senate control, 50-50, but effectively handed the majority to the Democrats, because of the tiebreaking vote of Vice President Kamala Harris, who serves as the body’s president. Mr. Ossoff’s term does not expire until 2027.“This is a personal decision, not a political one,” Mr. Perdue wrote on Twitter Tuesday.He added that he was “confident” anyone the Republicans nominated to run would defeat Mr. Warnock, and said, “I will do anything I can to make that happen.” It is not clear why Mr. Perdue made his about-face. His post announcing his decision not to run butted awkwardly against his last one, posted on Feb. 16, that read, “Bonnie and I are considering running in 2022.”A message to his spokesman was not immediately returned.Ms. Loeffler and Mr. Perdue tethered their electoral prospects in the Jan. 5 runoffs closely to former President Donald J. Trump. In his statement on Tuesday, Mr. Perdue echoed Mr. Trump’s false claims of widespread voter fraud in the state and he called on Republican officials in Georgia to change state laws and election rules “so that, in the future, every legal voter will be treated equally and illegal votes will not be included.”State election officials have repeatedly said that illegal voting had no impact on the outcome of either the November general election or the January runoff. But in his statement last week, Mr. Perdue suggested he was the real winner of his race, even though his victory over Mr. Ossoff last fall fell below the state’s 50 percent threshold, triggering the runoff in January.